Theorem csbiebt 3875

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3879.) (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbiebt	⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebt

Step	Hyp	Ref	Expression
1		elex 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		spsbc 3751	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	adantr 486	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → 𝐴 ∈ V)
5		biimt 363	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
6		csbeq1a 3860	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	eqeq1d 2762	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 7	bitr3d 284	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	8	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
10		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ V
11		nfnfc1 2925	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝐶
12	10, 11	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ Ⅎ𝑥𝐶)
13		nfcsb1v 3870	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
15		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥𝐶)
16	14, 15	nfeqd 2932	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
17	4, 9, 12, 16	sbciedf 3780	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
18	3, 17	sylibd 242	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
19	13	a1i 11	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
20		id 23	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥𝐶)
21	19, 20	nfeqd 2932	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
22	11, 21	nfan1 2236	. . . . . 6 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
23	7	biimprcd 253	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
24	23	adantl 487	. . . . . 6 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → (𝑥 = 𝐴 → 𝐵 = 𝐶))
25	22, 24	alrimi 2249	. . . . 5 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
26	25	ex 418	. . . 4 ⊢ (Ⅎ𝑥𝐶 → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
27	26	adantl 487	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
28	18, 27	impbid 215	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
29	1, 28	sylan 592	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  Vcvv 3450  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847

This theorem is used by:  csbiedf  3876  csbieb  3877  csbiegf  3879